Let S the top of the tripod, A,B,C the 3 feet with AB=BC=CA.
Let P the projection of S in the plane ABC
Let M the middle of segment BC
Consider a small diminution S'S'' of the leg's length SA
then
S move to S' on the plane MAS on a circle of centre M and ray MS.
--> It is a 2D problem!
let S'S'' be on the line AS' with AS'' = AS
As S'S'' is small you can consider SS' as a segment pependicular to MS and
SS'' a segment perpendicular to AS.
You have 2 beautiful similar triangle SS'S'' and SAP with S'S'' the known
leg's error and SS' the moving in a known direction..
--> No trigo, no Pythagore.!!
With S'S'' small you can add the error vector from the 2 other legs. The
second order moving errors are negligible.
Of course you need Mathematica to transform the coordonate system SABC to
your favorite one with 3x3 matrix. It will do the trigo for you
Similar reasoning for small displacment of A, B and C
Simple is beautiful.
Pierre Ravussin
>I am working on a solid geometry problem that is fairly straightforward
>but algebraically hideous. I could write MMA code to deal with it, but
>I was wondering if anybody knew of any packages or shortcuts.
>
>The application deals with what is known as a "kinematic mount"
>for precision instrument placement in optics. The idea is this:
>suppose you position an object that is mounted on a tripod. If
>each leg is the right length and is placed right where you want it
>you have perfect placement. The problem is computing the error
>in placement if a leg is the wrong size or is placed in the wrong spot.
>
>Does anybody know of some neat tricks for doing solid geometry?
>That is, can MMA do the three dimensional construction without
>the brute force approach of exhaustive application of Pythagoras
>and trig?
>
>All ideas appreciated! Thanks,
>
>Keith
>
>clayka at plu.edu
>clay at phys.washington.edu
>
>
>
>
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